On the supremum of random cusp forms
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908500008894464 |
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| author | Huang, Bingrong Lester, Stephen Wigman, Igor Yesha, Nadav |
| author_facet | Huang, Bingrong Lester, Stephen Wigman, Igor Yesha, Nadav |
| contents | A random ensemble of cusp forms for the full modular group is introduced. For a weight-$k$ cusp form, restricted to a compact subdomain of the modular surface, the true order of magnitude of its expected supremum is determined to be $\asymp \sqrt{\log{k}}$, in line with the conjectured bounds. Additionally, the exponential concentration of the supremum around its median is established. Contrary to the compact case, it is shown that the global expected supremum, which is attained around the cusp, grows like $k^{1/4}$, up to a logarithmic factor. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_16813 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the supremum of random cusp forms Huang, Bingrong Lester, Stephen Wigman, Igor Yesha, Nadav Number Theory Classical Analysis and ODEs Probability 11F11, 60G60 A random ensemble of cusp forms for the full modular group is introduced. For a weight-$k$ cusp form, restricted to a compact subdomain of the modular surface, the true order of magnitude of its expected supremum is determined to be $\asymp \sqrt{\log{k}}$, in line with the conjectured bounds. Additionally, the exponential concentration of the supremum around its median is established. Contrary to the compact case, it is shown that the global expected supremum, which is attained around the cusp, grows like $k^{1/4}$, up to a logarithmic factor. |
| title | On the supremum of random cusp forms |
| topic | Number Theory Classical Analysis and ODEs Probability 11F11, 60G60 |
| url | https://arxiv.org/abs/2508.16813 |